I play the Two Knighs Defense as Black vs. 1. e4 e5 2. Nf3 Nc6 3. Bc4, in the spirit of counterattack. What would you suggest I play as Black vs. the Ruy Lopez?
There's a simple reason for which 3...a6 and 3...Nf6 are by far the most common replies to the Ruy Lopez: they're the most solid for black. It's very hard to force white into playing something dynamic without comprimising your own position. Of course, you can try something like the Schliemann Defence with 3...f5, though objectively this is dubious for black. However, assuming you aren't playing against titled players, it can still be worth trying if you have good preparation and if you trust that your tactical ability is better than your opponent's. Another option instead of the Schliemann is the so-called "Siesta variation", which features a similar f5 pawn break, though black scores better here than in the Schliemann.
[FEN ""] 1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 d6 5. c3 f5
This leads to very dynamic play. GM Laurent Fressinet used this in 3 different games at the World Blitz Championship 2017 against other GMs, and won all 3 of them. GM Valeri Yanderimov also has many games in this variation, with mixed results.
Alternatively, if you want something different, you can always try the Cozio defense, both the standard and the deferred variations. This is more solid, though players at lower levels are less likely to be prepared in this opening, and the resulting positions can be surprisingly dynamic with potential pawn storms and black having a strong bishop on g7. The answer to this question provides good insight into the differences between the direct and deferred versions. Of course, the dynamic character of the Cozio depends largely on what white decides to play, and thus you still have to be ready for more quiet or more drawish positions.